Trees of integral triangles with given rectangular defect

AbstractThe rectangular defect of a triangle with side lengths a, b and c is a2+b2−c2 where a,b≤c. For a given integer d we examine the set PIT(d) consisting of all primitive integral triangles with rectangular defect equal to d. There are simple transformations τ1, τ2 and τ3 which produce new elements of PIT(d) from any triangle with defect d. They determine a partial ordering on PIT(d) in which applying any τi moves upward. We will show that the poset PIT(d) has finitely many components and that each of these components is isomorphic to one of two rooted trees T or T˜ (where T is the regular rooted tree of valence three and T˜ is a subtree of it). It follows that the minimal elements of PIT(d) form a finite set from which any triangle in PIT(d) can be uniquely obtained by applying a finite sequence of the τi’s.In order to prove these statements we will analyze a larger poset Σ(d) which contains copies of both PIT(d) and its inverse −PIT(d) as subposets. The elements of Σ(d) are equivalence classes of solutions to the equation x12+x22+x32−2x2x1−2x2x3=d. The key result will assert that the complement of ±PIT(d) in Σ(d) is a finite poset, denoted by Core(d). The proof of this key result is very different according to whether d is nonpositive (the obtuse case) or d is positive (the acute case), and the two cases must be analyzed separately. In the obtuse case we will see that the components of Core(d) are singletons while in the acute case they are poset segments or poset circuits (these are the finite connected posets in which each element has at most two neighbors). For all values of d the analysis of Σ(d) will produce algorithms for constructing both Core(d) and the minimal elements of PIT(d)

Single-spin asymmetries: the Trento conventions

During the workshop "Transversity: New Developments in Nucleon Spin Structure" (ECT*, Trento, Italy, 14-18 June 2004), a series of recommendations was put forward by the participants concerning definitions and notations for describing effects of intrinsic transverse momentum of partons in semi-inclusive deep inelastic scattering.Comment: 6 pages, 1 figure. Outcome of discussions at the workshop "Transversity: New Developments in Nucleon Spin Structure" (ECT*, Trento, Italy, 14-18 June 2004). Footnote 2 adde

Happily entangled: prediction, emotion, and the embodied mind

Recent work in cognitive and computational neuroscience depicts the human cortex as a multi-level prediction engine. This ‘predictive processing’ framework shows great promise as a means of both understanding and integrating the core information processing strategies underlying perception, reasoning, and action. But how, if at all, do emotions and sub-cortical contributions fit into this emerging picture? The fit, we shall argue, is both profound and potentially transformative. In the picture we develop, online cognitive function cannot be assigned to either the cortical or the sub-cortical component, but instead emerges from their tight co-ordination. This tight co-ordination involves processes of continuous reciprocal causation that weave together bodily information and ‘top-down’ predictions, generating a unified sense of what’s out there and why it matters. The upshot is a more truly ‘embodied’ vision of the predictive brain in action

A global fit of top quark effective theory to data

In this paper we present a global fit of beyond the Standard Model (BSM) dimension six operators relevant to the top quark sector to currently available data. Experimental measurements include parton-level top-pair and single top production from the LHC and the Tevatron. Higher order QCD corrections are modelled using differential and global K-factors, and we use novel fast-fitting techniques developed in the context of Monte Carlo event generator tuning to perform the fit. This allows us to provide new, fully correlated and model-independent bounds on new physics effects in the top sector from the most current direct hadron-collider measurements in light of the involved theoretical and experimental systematics. As a by-product, our analysis constitutes a proof-of-principle that fast fitting of theory to data is possible in the top quark sector, and paves the way for a more detailed analysis including top quark decays, detector corrections and precision observables.Comment: Additional references and preprint code added. Minor error in generation of plots fixed, no conclusions affecte

The size of the X-ray emitting region in SWIFT J2127.4+5654 via a broad line region cloud X-ray eclipse

We present results obtained from the time-resolved X-ray spectral analysis of the Narrow-Line-Seyfert 1 galaxy SWIFT J2127.4+5654 during a ~130 ks XMM-Newton observation. We reveal large spectral variations, especially during the first ~90 ks of the XMM-Newton exposure. The spectral variability can be attributed to a partial eclipse of the X-ray source by an intervening low-ionization/cold absorbing structure (cloud) with column density N_H = 2.0^{+0.2}_{-0.3}e22 cm^-2 which gradually covers and then uncovers the X-ray emitting region with covering fraction ranging from zero to ~43 per cent. Our analysis enables us to constrain the size, number density, and location of the absorbing cloud with good accuracy. We infer a cloud size (diameter) of $D_c < 1.5e13 cm, corresponding to a density of n_c > 1.5e9 cm^-3 at a distance of R_c > 4.3e16 cm from the central black hole. All of the inferred quantities concur to identify the absorbing structure with one single cloud associated with the broad line region of SWIFT J2127.4+5654. We are also able to constrain the X-ray emitting region size (diameter) to be D_s < 2.3e13 cm which, assuming the black hole mass estimated from single-epoch optical spectroscopy (1.5e7 M_sun), translates into D_s < 10.5 gravitational radii (r_g) with larger sizes (in r_g) being associated with smaller black hole masses, and viceversa. We also confirm the presence of a relativistically distorted reflection component off the inner accretion disc giving rise to a broad relativistic Fe K emission line and small soft excess (small because of the high Galactic column density), supporting the measurement of an intermediate black hole spin in SWIFT J2127.4+5654 that was obtained from a previous Suzaku observation.Comment: 8 pages, 7 figures, accepted for publication in MNRA

Results from TopFitter

We discuss a global fit of top quark BSM couplings, phrased in the model-independent language of higher-dimensional effective operators, to the currently available data from the LHC and Tevatron. We examine the interplay between inclusive and differential measurements, and the complementarity of LHC and Tevatron results. We conclude with a discussion of projections for improvement over LHC Run II.Comment: 5 pages, 4 figures, proceedings of the 9th International Workshop on the CKM Unitarity Triangle, 28 November - 3 December 2016, Tata Institute for Fundamental Research (TIFR), Mumbai, Indi